One of the first and most profound questions humanity has ever collectively asked is: Are we alone? While there are many ways scientists have approached answering this question, this method involves determining the geometry of an object, and in turn whether it is naturally occurring or not, solely based on the slope of the light curve during transit. The purpose of this study is to analyze the results of a simulation for the transitory light curves of squares and triangles in order to predict how the slope would vary when the object being observed is a geometrically shaped extraterrestrial megastructure (EM) rather than a spherically shaped planet. At this scale of observation, all naturally occurring objects would accrete into a roughly spherical shape, so the results of this model could help differentiate between naturally occurring and constructed celestial bodies. This model, and its results, could allow future astronomers to sift through the light curve of hundreds of thousands of transiting exoplanets to find an unnatural orbiting object based solely on its geometric features. This study includes predictions for the light curve of a rectangular prism with multiple limb darkening scenarios. Most approaches for detecting megastructures rely on measuring extreme variable brightness, but these large swings can often be explained by brown dwarfs forming a binary system or a large and inconsistent dust cloud. Building a model in Python to analyze the slope of the light curve of geometric shapes allows for not only a much broader scheme of detection but a higher certainty after detection, potentially helping resolve the anomalous observations of Tabby’s star and many other stars in the future. Predicting multiple scenarios for the light curve of a geometric celestial body allows scientists to better understand the slopes of the light curves of potential extraterrestrial megastructures, and estimate the probability of a false positive, paving the way for future research.

Introduction

The main method astronomers use to detect planets outside of our solar system, known as exoplanets, is the transit method. Many other methods are also used, such as astrometry, radial velocity, gravitational microlensing and direct imaging, but the transit method is by far the most popular [1]. The transit method involves looking at the luminosity of a star over time and detecting small dips in the brightness in a periodic fashion due to an exoplanet passing between the observer and the star. This method only works for a fractional number of cases as the planetary plane must be what astronomers call “edge-on”, meaning its planetary plane lines up with our perspective and scientists can actually observe a transit [2]. This configuration is only present in roughly 1% of all stellar systems so scientists typically monitor tens of thousands or hundreds of thousands of systems to see these effects [3]. The probability of a planet at 1 AU (the average distance between the earth and the sun) being edge-on is even lower, at 0.47% [4]. When the time it takes the transit to occur is estimated and the percentage of the star's light that is blocked is also approximated, a radius of the planet can be derived. The total change in luminosity is not only dependent on the radii of both the orbiting body and the star, but is also heavily impacted by the orbital radius of the body, as that is what determines the relative apparent radius. The equation below can be used to estimate the total change in flux of a star based on the relative apparent radius of the orbiting body, where F denotes flux, Rp is the radius of the planet and R* is the radius of the star [5].

When this is combined with the radial velocity method, scientists can determine the mass of the planet and in turn the density to figure out whether the planet is rocky or gaseous. However, this method mostly discovers large bodies that are very close to the star with a short orbital period because those are the easiest to detect and confirm with repeated observations. The larger the apparent radius of the planet relative to that of the star the greater the dip in the brightness will be, making it easier to detect. The shorter the orbital period, the faster astronomers observe multiple transits and confirm the existence of a planet.

All celestial objects roughly 400-600 kilometers in diameter will naturally be shaped into a sphere by their own gravity. All objects over 1,000 kilometers in diameter must be roughly spherical or further collapse will take place [6]. As our exoplanet-hunting telescopes have only a high enough angular resolution to detect the signatures of planets over 4,000 kilometers in diameter [7], all planets detected by Kepler and other telescopes using the transit method have a spherical shape. This spherical shape creates a very predictable “light-curve”, meaning the dip in the brightness always has a similar shape and slope (assuming parameters such as orbital period and radius remain constant). When all variables of a system are known, such as the orbital radius, luminosity, limb darkening coefficient among others, a very accurate model of that light curve can be created.

Civilizations could create different, possibly geometric, shapes, that theoretically could have a very different light curve simply based on the shape of the orbiting object. At first this idea seems preposterous: How could a civilization create a structure that large? The universe is 13.7 billion years old and it is reasonable to expect that, as life could have theoretically developed at any point roughly 5 billion years into the existence of the universe, alien life would be far more technologically advanced [8]. Humans have been a scientific civilization for roughly 400 years. It was 400 years ago when humans first started using science to drive inquiry, investigation and innovation. Extraterrestrial civilizations, if they do exist, are most likely thousands if not millions of years older than human civilizations, and as such could have technology much more advanced than we do [9].

Frank Drake, the namesake of the famous “Drake Equation,” shares the viewpoint of many modern astronomers; we likely share the galaxy with dozens, if not hundreds, of extraterrestrial civilizations [10]. This is based on the idea that even if the odds of life developing on a planet are astronomically low, the sheer number of stars in our galaxy virtually guarantee that in one of those star systems, intelligence life has risen.

Astronomers typically use the Kardashev scale when assessing the complexity of a civilization, civilizations range from Type I to Type III, based on the total energy consumption. Many of these civilizations are likely Type II, meaning they are harnessing the power of their entire host star [11]. In order to accomplish this they likely require an array of structures that encompass their star [12]. This type of structure is commonly referred to as a “Dyson Sphere” [13]. Some of these may have varying sizes and some could easily become large enough to have their light curve observed through a large enough telescope. Furthermore, if a civilization wanted a consistent way of alerting the rest of the galaxy to their existence, building an EM is a viable option. Type III civilizations consume orders of magnitude more than Type II, and thus must control entire galaxies. Based on this assumption, astronomers can effectively rule out the possibility that a Type III civilization exists within our galaxy, as some indication of them would have been detected by now given current technology.

One key concept when trying to understand transits is contact points. This image, courtesy of ESA’s Transit of Venus blog page on Transit Terminology illustrates the idea quite well. The 1st contact point is when the planet begins to occlude the exterior of the star, and the 2nd contact point is when it is fully transiting and the entirety of the orbiting object is occluding light from the star. The 3rd and the 4th contact points are the reverse, occurring towards the end of the transit duration [14].

For objects of equal “cross sectional area,” relative to their star, the total brightness at maximum transit should be roughly identical. However, there is a potential for a difference in the light curve between the 1st and 2nd contact points, and 3rd and 4th contact points. More specifically, the slope between those points may be significantly different so that a model for these scenarios could help scientists differentiate between the two consistently, and that is the basis of the study.

There is a complex list of the many forms an alien megastructure or system of structures may take and that is why an equally complex computer model needs to be created to predict their transit. The model built makes predictions for the light curve of a geometric prism and a sphere (for control purposes), this study analyzes the differences in those computed light curves. The question this research addresses is: Can extraterrestrial megastructures be detected solely based on their geometric features? The hypothesis of this study is that given a strong understanding of the nature of the system, meaning all factors and variables that impact the light curve are well understood, along with modern telescopic technology, astronomers are able to detect a geometric EM.

Methods

Fundamental Mechanics

Simulations of an idealized transit system were carried out for a variety of object shapes and star brightness profiles. The simulation was created in Python, a programming language particularly suited for science research because of the abundance of data analysis packages and libraries. The model is two dimensional, and the relative apparent surface area is the driving factor in the change in flux. The model’s adjustable parameters are the total number of points in the planet and the star. The control group is the circle and the experimental group is the square and triangle. In the simulation, a star is placed in a fixed point in two dimensional space. The motion of a celestial body between the viewer and the center of the star is plotted from 1st contact to 4th contact, and the intersection points are then mapped and weighted with intensity values. To effectively compare the two graphs, both the control object (a circle), the square and triangle have roughly the same number of total points. The parameters are rounded for performance reasons (see end of calculation section for explanation). The total luminosity is calculated at each movement and is graphed as a function of time. The total number of points in each graph varies depending on whether it is a square, a circle or triangle because the side length and the diameter vary. The area between the first and second contact point, the main area of interest for this investigation, remains the same. This allows for a juxtaposition of the two slopes, which ultimately leads to the conclusion of the study. As spheres naturally accrete and prisms do not, this concurs with our understanding of planetary formation.

Stellar Properties

For the purposes of this model, the “star” is a set of points arranged in a circle. The object creating the transit is a set of points arranged in a square, circle or triangle depending on whether it is an EM or planet. The object is moved over the star, and all intersection points are found for every position and subtracted from the total points in the star. The light curve is then made, by finding the ratio of the non occluded points to the total, by subtracting the number of intersecting points from the total and subtracting by the total number of points.

For performance reasons, the star is made to have few total points, 9852 (a 2D radius of 56 points), but the total perceived number of points varies due to the varying intensity mapped by the limb darkening function (see limb darkening section for more information). Despite this apparent oversimplification, this is a reasonable approximation because astronomers often observe few photons from stars. The most high powered telescopes typically observe around ten times the number of photons as points provided by this model.

This graph above shows the number of intersecting points vs time for a square occluding a total of 2% of the star. Without a changing intensity based on the distance from the center of the star, the transit curve would appear to be the inverse of this graph, as light curve is the non occluded points over the total. Even with limb darkening, explained below, the slope of the intersection function for each object remains consistent, meaning the only variation is caused entirely by the geometric features of the different shapes.

Limb Darkening/Nonlinear Optical Variation Approximations

One effect that would otherwise not be accounted for in a model if the change in brightness from near full transit to full transit was approximated as a constant would be “limb darkening.” Limb darkening is an optical phenomenon where the visual edges of the sun or any star will appear darker than the center. This is because the outer layers of the star are being viewed far less directly as they are traveling through a significantly larger amount of gas, and the luminosity decreases [15]. Some stars have stronger limb darkening function than others, depending on the density of gas in the photosphere of a star. A strong, or aggressive, limb darkening model refers to a function that significantly differentiates between photon emission at the limb and the center. Conversely, a weak, or limited, limb darkening function has a much smaller difference between photon intensity and the edge of a star and the center. A basic understanding of this non-constant luminosity distribution near full transit is necessary for modelling an accurate slope, as that is precisely the area of interest of this investigation.

Astronomers using this model should use a particular star’s limb darkening function in order to correctly juxtapose the light curves produced by the simulation. As the model is downloadable (see additional information section), it is suggested astronomers calibrate the limb darkening function to their star for the most accurate results.

There should be little difference in a light curve between a square, triangle and circle if they have the same 2D area on the star when at full transit, but a large difference that could potentially be detected is at the near transit phase (between the 1st and 2nd contact points). This is because full transit will be achieved at different times and at different rates depending on the geometry of the object. If the slope is even slightly affected by the presence of another non constant force apart from the apparent geometry of the object, it must be incorporated in order to produce a curve that accurately reflects what data would look like coming from an EM. Typical Kepler Space Telescope observations are between 420-890 nanometers, wavelengths more sensitive to limb darkening, heightening the importance of developing a reliable limb darkening model [16]. To adjust for this effect, the central pixels have been weighted relative to external pixels, using the equation below [17].

The first contact point is defined 90°, it then decreases to the center point which is 0° and then it increases again to 90°. This has the effect of making the exterior points weighted 60% less than the center point. This is consistent with most strong limb darkening models for sun like stars. When scientists are trying to differentiate between a megastructure and a planet, the limb darkening model must be known very thoroughly. This basic model allows one fairly common scenario to be used to contrast what the light curve of an EM and that of a planet would look like.

This graph shows the output of the limb darkening function (0.4 + 0.6cos?), as the angle, and therefore the time, changes. Every point on the light curve is calculated by taking the non occluded points (the inverse of the intersecting points graph) and multiplying by the weighting for each point to find total luminosity. In this way, the light curve is a combination of the limb darkening function and occluded points graph. As such, it is expected that the weaker the limb darkening function, the more distinguishable the light curves will be, as the limb darkening function remains consistent across both the megastructure and the planet and the intersection function is the only area of variation between the two objects’ light curve.

To increase the flexibility of this model for real world application the light curves will also be calculated with uniform brightness and no limb darkening. Most stars exist in a range between an aggressive limb darkening function and none at all, so analyzing these two scenarios provides a convenient way to compare overall how the light curves differ.

Tidal Locking

Another assumption made is that the object is tidally locked, and therefore does not appear rotate on an axis when observed from the reference frame of earth. Based on the sheer convenience of assuming an EM to behave as tidally locked and the benefits it would have, namely gravity-gradient stabilization, especially at the small radius the object is presumed to be orbiting at, this premise is fair [18].

In addition, as the time of transit relative to the orbital period is incredibly small, over the course of a single transit a minuscule arc-second shift in the rotation would not be expected during a typical observation. If, however, an object was observed repeatedly for confirmation purposes, and was not tidally locked, one could expect a curve of varying slope over the course of those multiple observations.

Inclination

As discussed in the introduction, in order for the transit method to be applicable, the object must be “edge on,” meaning that relative to earth it has very low inclination. For the purpose of this model the inclination, meaning the degree variation from the reference plane to the orbital plane, is 90°. The amount of variation that could occur depends heavily on the semi-major axis of the orbiting object, so the extent to which this assumption is reasonable cannot be precisely determined [19]. However, even with objects whose orbital period is provided in mere days and not years, the highest inclinations recorded using the transit method are at most 5° from the reference plane [20]. As 5° is such a small deviation from the reference plane, allowing fluctuation in inclination would create divergences in the modelling that would over-complicate an already complex model. Comparison would also become needlessly more convoluted as there would be dozens of scenarios to analyze.

Transit Duration

In order to reduce unnecessary complications, the transit duration of all objects is identical (specifically 115 data points). As the purpose of this model is to compare light curves, adding variability in transit duration would impair the ability to effectively differentiate the different light curves. Furthermore, the area of interest is between the 1st and 2nd contact points, so the total duration is more or less irrelevant for this investigation given that the area between the 3rd and 4th contact points is exactly symmetrical.

Also, the model is simplified as two dimensional, and as such the movement of the body approximated as orthogonal to field of view rather than the star. Changes in the components of the velocity throughout the duration of the transit could change slope of the light curve in a way that is not adjusted for in this model. This effect is constant, and also more or less irrelevant if the semi-major axis is sufficiently large, so it has little to no effect on the conclusion of the study.

Data

Light Curve Comparison

The circle (planet), square and triangle (square and triangle being part of the experimental group) will have six graphs associated with it. Two graphs will be comparing the two light curves with both objects obscuring roughly 1% of the total light, two with 2% and another pair with 20%. Graphs of the same parameters will be created without limb darkening both, for comparison and to ensure the results of this study are as complete as possible. A 1% decrease in brightness is typical to most large “Hot Jupiters” and 2% is generally on the higher end for the largest gas giants in close orbits around their stars. 20% is in the range of a binary system that has a “brown dwarf,” a failed star much larger than Jupiter but significantly smaller than stars. Another potential cause of a change in flux that large is a thick debris disk, as hypothesized with Tabby’s star [21]. This scenario is improbable but entirely plausible, and the large light curve would already indicate a megastructure, so modelling a large body helps predict what scientists would observe in these key circumstances. Given the already immense scale of these objects, modelling these situations helps to rule out the possibility of a brown dwarf transit and potentially alert astronomers to the possible presence of an EM.

The control object is a spherical planet of a similar “surface area” to the star as the geometric prism, but in the shape of a naturally occurring planet. This is done for a concise comparison of the slopes of the two light curves. The data included is the slope between the first and second contact point of the transit, emphasizing the portion containing the variation as all objects are designed to have the same total cross sectional area and in turn the same minimum luminosity. The complete light curve is also included for reference, but the area between the third and fourth contact points, which is symmetrical to the area between the first and second contact points, is cut off from some of the graphs to ensure that all of the data sets have the same total number of data points. As the section of the graph that is vital to this study is between the first and second contact points, a difference in terms of the total number of points in the graph is not material to the result. The y-axis of the graph is the luminosity, calculated as the sum of the unobstructed points weighted by the limb darkening model. The x-axis is time. Since the control object and the prism are of similar size and the number of unobscured points is the same, the existence of a limb darkening function results in a different trough of the luminosity curve. The time of the 2nd contact point also varies with the total transit time, so calculations may not be visually consistent with the graphs cutoff point. The percent difference calculation refers to the graph of the luminosity between the 1st and 2nd contact points, which as previously stated do not always occur at the exact same time (refer to the calculations for specific values). The 2nd contact point is defined by finding when the light curve levels off at a certain plane. The point is abundantly clear for the weak limb darkening function graphs and less apparent for the limb darkening light curves as shown in the section below. The second contact point can be noticed, where there is a rapid change in the slope on the limb darkening graphs prior to full transit. The graphs were constructed using Microsoft Excel, but the values were directly from the simulation.

With Limb Darkening

Percent Differences by Occlusion

Square

1%: 5.32

2%: 1.58

20%: 0.55

Average: 2.48

Triangle

1%: 2.5

2%: 0.81

20%: 32.42

Average: 11.91

Without Limb Darkening

Percent Differences by Occlusion

Square

1%: 0.55

2%: 1.22

20%: 13.76

Average: 5.12

Triangle

1%: 4.82

2%: 0.85

20%: 9.82

Average: 5.12

Calculations

Calculations were made for the side length of the megastructure and the radius of the planet as well as the radius of the star for each observation, they are listed below. Bolded numbers represent the parameters to enter into the program and values in italics are the percent difference.

The numbers were rounded, as adding additional numbers after the decimal would force the program to scan a hundred times the number of points for every decimal of precision, which slows the program significantly. As the deviation in the slope of the light curve is the most important piece of data and differences in timing before reaching full transit are accounted for, rounding the parameters and therefore having a difference between the 2nd and 3rd contact points has little effect on the conclusion. The total luminosity of the star is also not a full 10,000, as seen in the graph, this is due to limb darkening making the exterior regions weighted 60% less than the center, which makes the limb weighted at 0.4, and rounding of the star’s radius. Differences in the bottom of the light curve are likely due to rounding of calculations and the limb darkening model changing the luminosity at different rates based on the geometry of the objects by small margins.

Discussion/Conclusion

This investigation yielded an unexpected result. The light curve is heavily dependent on the limb darkening model as well as the geometry of the object. Based on the data collected from this simulation, the transitory light curve of an extraterrestrial megastructure (EM) would be sufficiently different from an exoplanet to enable scientists to distinguish between the two based on their geometric features alone. The percent difference of the planet and the EM, which serves as an aggregate for the likelihood the two can be reasonably differentiated, indicates that there is a possibility scientists will be able to use this model to determine whether the object in question is a megastructure or a planet. A square with weak limb darkening at 20% occlusion has a 13.76% separation from the control and an average of 5.12% across the three occlusion ratios. Also, as it has a 5.32% difference, the contrast between a megastructure and a planet with strong limb darkening at 1% occlusion is very significant. The triangle had even higher percent differences, with 32.42% with limb darkening and 9.82% difference without limb darkening, both at 20% occlusion. These very large percent differences are easily within the ability of current technology to differentiate between them. However, to increase the certainty of a detection of an EM, astronomers should compare the entirety of the light curve along with adjusting based on the specific stars limb darkening function. Contextualizing the light curve by looking at the entire curve, and not solely on the area between the 1st and 2nd contact points, would prove to be key given the unanticipated results of research.

With a strong understanding of the nature of the system, meaning variables such as the radius, orbital radius, luminosity, limb darkening coefficients, eccentricity, and inclination are known accurately, scientists will be able to use this model to differentiate between a geometrically shaped EM and a planet. Understanding the specific limb darkening model is crucial, as the data collected reveals that a strong limb darkening model will lead to a greater difference in the two light curves between the 2nd and 3rd contact points, whereas limited limb darkening will cause the greater difference to occur towards edges of the star. When approached with an abnormal reduction in luminosity, astronomers can use this model to estimate the probability that the anomaly in a light curve signals the existence of an extraterrestrial megastructure based solely on geometric appearance of the object. The results also determined that the weaker the limb darkening function, the more likely scientists will be able to discriminate between the two bodies. Of course, this model portrays only one of the many ways that an EM can be detected. This method is subtle but sound, as in an ideal scenario where the parameters of an orbiting body are known very precisely, the accuracy of modern telescopic observations may allow scientists to see if the object is irregularly shaped.

As telescopes become more advanced, and their observations more exact, the probability that this model will be applicable increases dramatically. Given the conservative estimate of 10,000 Type II civilizations in the galaxy, and the 1% probability that any system would be edge on relative earth, it is anticipated that 100 of such objects should be detectable out of the 400 billion stars in the galaxy. The task to identify candidate objects seems daunting, but as more exoplanet hunting telescopes are launched, each continuously monitoring hundreds of thousands of stars, the chance of coming across an object such that a model to describe their light curves may be useful becomes more likely. Taking this approach to the detection of alien life is convenient and definitive, contributing to the comprehensive approach scientists are taking to finally find extraterrestrial intelligence. This area of research adds a unique perspective to the efforts on the part of contemporary astronomers to answer the question of the nature of life in the universe.